Not that the optimal solution \(P^\epsilon \)  corresponding to \(\mathrm {L}^{\epsilon }\)  can also be differentiated w.r.t. any of the relevant inputs, using the implicit function Theorem [1]}, as proposed in OTT-JAX [2]}, but this is not used in this paper because we rely on Danskin's Theorem [3]} (a.k.a Envelope Theorem) to differentiate \(\mathrm {L}^{\epsilon }\)  w.r.t. \(\mathbf {C}\) .
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